# Quantitative Methods II

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Quantitative Methods II
Dummy Variables & Interaction Effects Edmund Malesky, Ph.D., UCSD

The Homogeneity Assumption
OLS assumes all cases in your data are comparable x’s are a sample drawn from a single population But we may analyze distinct groups of cases together in one analysis Mean value of y may differ by group

Qualitative Variables
These group effects remain as part of the error term If groups differ in their distribution of x’s, then we get a correlation between the X variables and the error term Violates assumption: cov(Xi, ui)=E(u)=0 Omitted Variable Bias!

Testing for Differences Across Groups (p. 249-252)
The Chow Test: Is only valid under homoskedasticity (the error variance for the two groups must be equal). The null hypothesis is that there is no difference at all; either in the intercept or the slope between the two groups. This may be two restrictive in these cases, we should allow dummy variables and dummy interactions to allow us to predict different slopes and intercepts for the two groups. The Chow Test i.e. Testing for difference between males and females on academic performance. SSR1=Males only; SSR2=Females only SSRur=SSR1+SSR2 SSRP=SSRr=Pooling across both groups

Example: Democracy & Tariffs
But if Democracies are more likely to be in RTA’s, then pooling RTA and non-RTA states biases the coefficient Here we see that democracies have lower tariffs Here we see that states in Regional Trading Arrangements (RTA’s) have lower tariffs

Solution: The Qualitative Variable
Measure this group difference (RTA vs. Non-RTA) and specify it as an x This eliminates bias But we have no numerical scale to measure RTA’s Create a categorical variable that captures this group difference

The Qualitative “Dummy”
Create a variable that equals 1 when a case is part of a group, 0 otherwise This variable creates a new intercept for the cases in the group marked by the dummy Specifically, how would we interpret:

Democracy and Tariff Barriers

Graphical Depiction of a Dummy
x1 (could be continuous, categorical, or dichotomous)

Multiple Category Dummies
Dummy variables are a very flexible way to assess categorical differences in the mean of y We can use dummies even for concepts with multiple categories Imagine we want to capture the impact of global region on tariffs Regions: Americas, Europe, Asia, Africa

Warning! Do not fall into the dummy variable trap!
When you have entered both values of a dummy variable in the same regression. These two variables are linearly dependent. One will drop out.

Multiple Category Dummies
Create 4 separate dummy variables - 1 for each region Include all except one of these dummies in the equation If you include all 4 dummies you get perfect collinearity with the constant. The fourth dummy will drop out. Americas+Europe+Asia+Africa=1

Interpreting Multi-Category Dummies
Each coefficient compares the mean for that group to the mean in the excluded category Thus if: βhat2-βhat4 compare the mean tariff in each region to the mean in the Americas Mean in Americas is βhat0 An alternative strategy is to drop the constant and run all dummies, as discussed last week.

Dumb Dummies Dummy variables are easy, flexible ways to measure categorical concepts They CAN be just labels for ignorance Try to use dummies to capture theoretical constructs not empirical observations If possible, measure the theoretical construct more directly

Interaction Effects Dummy variables specify new intercepts
Other slope coefficients in the equation do not change OLS assumes that the slopes of continuous variables are constant across all cases What if slopes are different for different groups in our sample?

Interaction Effects: An Example
What if the effect of democracy on tariffs depends on whether the state is in an RTA?

Interaction Effects: An Illustration (Notice that democracy has been converted to a dummy as well for illustration purposes)

How Do We Estimate This Set of Relationships?
We begin with: Substituting for Βhat1, we get: Βhat1 Βhat2 Βhat3 In STATA, they will appear as regular coefficients

What Do These Coefficients Mean?

Interpreting the Interaction
Recall that: RTA is a dummy variable taking on the values 0 or 1

An Illustration of the Coefficients
Imagine we estimate:

Substantive Effects of Dummy Interactions
No RTA RTA Non- Democracy Βhat0 = 30 Βhat0 + Βhat3 = 20 Βhat0 + Βhat1 = 25 Βhat0 + Βhat1 + Βhat2 + Βhat3 = 14

Interactions with Continuous Variables
The exact same logic about interactions applies if Βhat1 depends on a continuous variable

Example: Democracy, Tariffs & Unemployment

Graphical Depiction of a Dummy/Continuous Interaction
x1 (could be continuous, categorical, or dichotomous)

What if a Variable Interacts with Itself?
What if Βhat1 depends on the value of x1? Then we substitute in as before: Curvilinear (Quadratic) effect is a type of interaction

More Complex Interactions
We can use this method to specify the functional form of βhat1 in any way we choose Simply substitute the function in for βhat1 , multiply out the terms and estimate Only limitations are theories of interaction and levels of collinearity

Examples of interaction effects from my own research

Governance and Economic Welfare

Land Use Rights Certificate
Predicted Number of Loans by Legal Status among Vietnamese Private Firms Land Use Rights Certificate Registered at DPI None Partial Full No 0.83 0.99 1.2 Yes 2.73 3.27 3.98

Predicted Probability of Provincial Division in Vietnam (By State Sector Output with Number of Cabinet Officials)

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